What Is Correlation?

Correlation analysis measures the strength and direction of a relationship between two numerical variables:

• Ranges from -1 to +1
• +1 indicates perfect positive correlation
• 0 indicates no correlation
• -1 indicates perfect negative correlation

The Pearson correlation coefficient (r) is defined as:

\[r = \frac{\sum_{i}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum_{i}(X_i - \bar{X})^2 \sum_{i}(Y_i - \bar{Y})^2}}\]

This can be simplified as:

\[r = \frac{\text{Covariance}(X, Y)}{s_X \cdot s_Y}\]

Where \(s_X\) and \(s_Y\) are the standard deviations of X and Y.

Example 16.1: Flipping the Bird

Interpretation: The correlation coefficient of r = 0.534 suggests that Nazca boobies who experienced more visits from non-parent adults as nestlings tend to display more aggressive behavior as adults. This supports the hypothesis that early experiences influence adult behavior patterns in this species.

Standard Error:

\(\text{SE}_r = \sqrt{\frac{1-r^2}{n-2}}\)

SE = 0.180

Need to be sure relationship is not curved - note below

Testing Assumptions for Correlation

As described in Section 16.3, correlation analysis has key assumptions:

1. Random sampling: Observations should be a random sample from the population
2. Bivariate normality: Both variables follow a normal distribution, and their joint distribution is bivariate normal
3. Linear relationship: The relationship between variables is linear, not curved

Let’s check these assumptions using the lion data from Example 17.1 Lion Noses:

Testing Assumptions for Correlation

As described in Section 16.3, correlation analysis has key assumptions:

1. Random sampling: Observations should be a random sample from the population
2. Bivariate normality: Both variables follow a normal distribution, and their joint distribution is bivariate normal
3. Linear relationship: The relationship between variables is linear, not curved

Let’s check these assumptions using the lion data from Example 17.1 Lion Noses:

Correlation: Important Considerations

The correlation coefficient depends on the range

• Restricting range of values can reduce the correlation coefficient
• Comparing correlations between studies requires similar ranges of values

Measurement error affects correlation

• Measurement error in X or Y tends to weaken observed correlation
• This bias is called attenuation
• True correlation typically stronger than observed correlation

Correlation vs. Causation

• Correlation does not imply causation
• Three possible explanations for correlation:
  1. X causes Y
  2. Y causes X
  3. Z (a third variable) causes both X and Y

Correlation significance test

• H₀: ρ = 0 (no correlation in population)
• H₁: ρ ≠ 0 (correlation exists in population)
• Test statistic: t = r / SE(r) with df = n-2

Simple Linear Regression Model

Simple linear regression models the relationship between a response variable (Y) and a predictor variable (X).

The population regression model \[Y = \alpha + \beta X + \varepsilon\]

Where:

• Y is the response variable
• X is the predictor variable
• α (alpha) is the intercept (value of Y when X=0)
• β (beta) is the slope (change in Y per unit change in X)
• ε (epsilon) is the error term (random deviation from the line)

The sample regression equation is:

\[\hat{Y} = a + bX\]

Where:

• \(\hat{Y}\) is the predicted value of Y
• a is the estimate of α (intercept)
• b is the estimate of β (slope)

Method of Least Squares: The line is chosen to minimize the sum of squared vertical distances (residuals) between observed and predicted Y values.

Simple Linear Regression Model

Simple linear regression models the relationship between a response variable (Y) and a predictor variable (X).

The population regression model is:

\[Y = \alpha + \beta X + \varepsilon\]

Where:

• Y is the response variable

• X is the predictor variable

• α (alpha) is the intercept (value of Y when X=0)

• β (beta) is the slope (change in Y per unit change in X)

• ε (epsilon) is the error term (random deviation from the line)

The sample regression equation is:

\[\hat{Y} = a + bX\]

Where:

• \(\hat{Y}\) is the predicted value of Y

• a is the estimate of α (intercept)

• b is the estimate of β (slope)

Method of Least Squares: The line is chosen to minimize the sum of squared vertical distances (residuals) between observed and predicted Y values.

\(\text{age} = 0.88 + 10.65 \times \text{proportion}_{black}\)

This means: - When a lion has no black on its nose (proportion = 0), its predicted age is 0.88 years - For each 0.1 increase in the proportion of black, age increases by 1.065 years - The slope (10.65) indicates that lions with more black on their noses tend to be older

Simple Linear Regression Model

The calculation for slope (b) is:
\[b = \frac{\sum_i(X_i - \bar{X})(Y_i - \bar{Y})}{\sum_i(X_i - \bar{X})^2}\]

Given: - \(\bar{X} = 0.3222\) - \(\bar{Y} = 4.3094\) - \(\sum_i(X_i - \bar{X})^2 = 1.2221\) - \(\sum_i(X_i - \bar{X})(Y_i - \bar{Y}) = 13.0123\)

b = 13.0123 / 1.2221 = 10.647

Intercept (a): \(a = \bar{Y} - b\bar{X} = 4.3094 - 10.647(0.3222) = 0.879\)

Making predictions:

To predict the age of a lion with 0.50 proportion of black on its nose:

\[\hat{Y} = 0.88 + 10.65(0.50) = 6.2 \text{ years}\]

Confidence intervals vs. Prediction intervals:

• Confidence interval: Range for the mean age of all lions with 0.50 black
• Prediction interval: Range for an individual lion with 0.50 black

Both intervals are narrowest near \(\bar{X}\) and widen as X moves away from the mean.

The test statistic is: \(t = \frac{b - \beta_0}{SE_b}\)

With df = n - 2 = 161 - 2 = 159

Interpretation:

The slope estimate is 0.033, indicating that log stability increases by 0.033 units for each additional plant species in the plot.

The p-value is very small (2.73e-10), providing strong evidence to reject the null hypothesis that species number has no effect on ecosystem stability.

R² = 0.222, meaning that approximately 22.2% of the variation in log stability is explained by the number of plant species.

This supports the biodiversity-stability hypothesis: more diverse plant communities have more stable biomass production over time.

Testing Regression Assumptions

linear regression has four key assumptions:

1. Linearity: The relationship between X and Y is linear
2. Independence: Observations are independent
3. Homoscedasticity: Equal variance across all values of X
4. Normality: Residuals are normally distributed

Let’s check these assumptions for the lion regression model:

Assume that error 𝞮 is \(e_i = y_i - \hat{y}_i\)

• normally distributed for each x_i

• has the same variance

• has a mean of 0 at each xi

Testing Regression Assumptions

linear regression has four key assumptions:

1. Linearity: The relationship between X and Y is linear
2. Independence: Observations are independent
3. Homoscedasticity: Equal variance across all values of X
4. Normality: Residuals are normally distributed

Let’s check these assumptions for the lion regression model:

Assume that error 𝞮 is - estimated as the residuals: \(e_i = y_i - \hat{y}_i\)

• ordinary lease square estimates a and b or slope and intercept to minimize the sum of the residuals squared or Mean Squared Error (MSE) as

\(\sum_{i=1}^{n} = (y_i - \hat{y}_i)^2\)

Testing Regression Assumptions

linear regression has four key assumptions:

1. Linearity: The relationship between X and Y is linear
2. Independence: Observations are independent
3. Homoscedasticity: Equal variance across all values of X
4. Normality: Residuals are normally distributed

Let’s check these assumptions for the lion regression model:

Testing Regression Assumptions

linear regression has four key assumptions:

1. Linearity: The relationship between X and Y is linear
2. Independence: Observations are independent
3. Homoscedasticity: Equal variance across all values of X
4. Normality: Residuals are normally distributed

Let’s check these assumptions for the lion regression model:

Testing Regression Assumptions

linear regression has four key assumptions:

1. Linearity: The relationship between X and Y is linear
2. Independence: Observations are independent
3. Homoscedasticity: Equal variance across all values of X
4. Normality: Residuals are normally distributed

Let’s check these assumptions for the lion regression model:

Simple Linear Regression Model

linear regression has four key assumptions:

1. Linearity: The relationship between X and Y is linear
2. Independence: Observations are independent
3. Homoscedasticity: Equal variance across all values of X
4. Normality: Residuals are normally distributed

If assumptions are violated: 1. Transform the data (Section 17.6) 2. Use weighted least squares for heteroscedasticity 3. Consider non-linear models (Section 17.8)